\newfont{\sqi}{cmssqi8}

\newcommand{\beq}{\begin{equation}}
\newcommand{\eeq}{\end{equation}}
\newcommand{\beqa}{\begin{eqnarray}}
\newcommand{\eeqa}{\end{eqnarray}}
\newcommand{\beqas}{\begin{eqnarray*}}
\newcommand{\eeqas}{\end{eqnarray*}}
\newcommand{\gap}{\hspace*{2em}}
\newcommand{\sgap}{\hspace*{1.2em}}
\newcommand{\bi}{\begin{itemize}}
\newcommand{\ei}{\end{itemize}}
\newcommand{\ben}{\begin{enumerate}}
\newcommand{\een}{\end{enumerate}}

\newcommand{\stroke}{\; | \;}

\newcommand{\bsli}{\begin{slide}{}}
\newcommand{\esli}{\end{slide}}

% \newcommand{\IR}{\makebox{\sf R \hspace{-13.5 pt} R \hspace{.3 pt}} }
% \newcommand{\IR}{{I\!\!R}}
\newcommand{\IR}{\makebox{\sf I \hspace{-9.5 pt} R \hspace{-7.0pt} }}
\newcommand{\R}{\IR}
\newcommand{\real}{\IR}
\newcommand{\IP}{\mbox{$I\!\!P$}}
\newcommand{\IN}{\mbox{$I\!\!N$}}
\newcommand{\IC}{\mbox{$C\!\!\!\!\raisebox{.75pt}{\mbox{\sqi I}}$}}
\newcommand{\bmat}{\left[ \begin{array}}
\newcommand{\emat}{\end{array} \right]}
\newcommand{\bvec}{\left( \begin{array}{c}}
\newcommand{\evec}{\end{array} \right)}
\newcommand{\ul}{\underline}
\def\half{{\textstyle{1\over 2}}}
\def\quarter{{\textstyle{1\over 4}}}

% my modification of the "itemize" environment for slitex
% (avoids the painfully distorted default)
\newcommand{\sitemize}{\begin{list}%
{$\bullet$}{\setlength{\itemsep}{0 pt}\setlength{\topsep}{0 pt}
\setlength{\partopsep}{0 pt}}}
\newcommand{\bsitemize}{\sitemize}
\newcommand{\esitemize}{\end{list}}
\newcommand{\insitemize}{\begin{list}%
{-}{\setlength{\itemsep}{0 pt}\setlength{\topsep}{0 pt}
\setlength{\partopsep}{0 pt}}}
\newcommand{\einsitemize}{\end{list}}

\newcommand{\sitemizeii}{\begin{list}%
{\bf --}{\setlength{\itemsep}{0 pt}\setlength{\topsep}{0 pt}
\setlength{\partopsep}{0 pt}}}
\newcommand{\bsitemizeii}{\sitemizeii}

\newcommand{\cA}{{\cal A}}
\newcommand{\cB}{{\cal B}}
\newcommand{\cC}{{\cal C}}
\newcommand{\cD}{{\cal D}}
\newcommand{\cE}{{\cal E}}
\newcommand{\cF}{{\cal F}}
\newcommand{\cH}{{\cal H}}
\newcommand{\cI}{{\cal I}}
\newcommand{\cJ}{{\cal J}}
\newcommand{\cK}{{\cal K}}
\newcommand{\cL}{{\cal L}}
\newcommand{\cM}{{\cal M}}
\newcommand{\cN}{{\cal N}}
\newcommand{\cP}{{\cal P}}
\newcommand{\cQ}{{\cal Q}}
\newcommand{\cS}{{\cal S}}
\newcommand{\cT}{{\cal T}}
\newcommand{\cU}{{\cal U}}
\newcommand{\cV}{{\cal V}}
\newcommand{\cW}{{\cal W}}

\newcommand{\thetain}{\theta_{\rm in}}
\newcommand{\thetaout}{\theta_{\rm out}}

\newcommand{\D}{\Delta}
\newcommand{\Dx}{\Delta x}
\newcommand{\Dy}{\Delta y}
\newcommand{\Ds}{\Delta s}
\newcommand{\Dl}{\Delta \lambda}
\newcommand{\Dlbar}{\Delta \bar{\lambda}}
\newcommand{\lam}{\lambda}
\newcommand{\lambar}{\bar{\lambda}}
\newcommand{\lamhat}{\hat{\lambda}}
\newcommand{\Dr}{\Delta r}
\newcommand{\Dw}{\Delta w}
\newcommand{\Dp}{\Delta \pi}
\newcommand{\sigmin}{\sigma_{\rm min}}
\newcommand{\sigmax}{\sigma_{\rm max}}
\newcommand{\gammin}{\gamma_{\rm min}}
\newcommand{\gammax}{\gamma_{\rm max}}
\newcommand{\gambar}{\bar{\gamma}}
\newcommand{\cNi}{{\cal N}_{-\infty}}
\newcommand{\cNinf}{{\cal N}_{-\infty}}

\newcommand{\xB}{x_{\cB}}
\newcommand{\xN}{x_{\cN}}
\newcommand{\sB}{s_{\cB}}
\newcommand{\sN}{s_{\cN}}
\newcommand{\DxB}{\Delta x_{\cB}}
\newcommand{\DxN}{\Delta x_{\cN}}
\newcommand{\DsB}{\Delta s_{\cB}}
\newcommand{\DsN}{\Delta s_{\cN}}
\newcommand{\Dz}{\Delta z}
\newcommand{\Dlb}{\Delta \bar{\lambda}}
\newcommand{\lb}{\bar{\lambda}}
\newcommand{\Dgb}{D \bar{g}}
\newcommand{\lB}{\lambda_{\cB}}
\newcommand{\lN}{\lambda_{\cN}}
\newcommand{\DlB}{\Delta \lambda_{\cB}}
\newcommand{\DlBa}{\Delta \lambda_{\cB1}}
\newcommand{\DlBb}{\Delta \lambda_{\cB2}}
\newcommand{\DlN}{\Delta \lambda_{\cN}}
\newcommand{\DgB}{D {g}_{\cB}}
\newcommand{\DgN}{D {g}_{\cN}}
\newcommand{\gB}{{g}_{\cB}}
\newcommand{\gN}{{g}_{\cN}}


\newcommand{\Dxaff}{\Delta x^{\rm aff}}
\newcommand{\Dyaff}{\Delta y^{\rm aff}}
\newcommand{\Dzaff}{\Delta z^{\rm aff}}
\newcommand{\Dpaff}{\Delta \pi^{\rm aff}}
\newcommand{\Dsaff}{\Delta s^{\rm aff}}
\newcommand{\Draff}{\Delta r^{\rm aff}}
\newcommand{\Dwaff}{\Delta w^{\rm aff}}
\newcommand{\Dxcc}{\Delta x^{\rm cc}}
\newcommand{\Dpcc}{\Delta \pi^{\rm cc}}
\newcommand{\Dscc}{\Delta s^{\rm cc}}
\newcommand{\Drcc}{\Delta r^{\rm cc}}
\newcommand{\Dwcc}{\Delta w^{\rm cc}}
\newcommand{\muaff}{\mu_{\rm aff}}

\newcommand{\tx}{\tilde{x}}
\newcommand{\tl}{\tilde{\lambda}}
\newcommand{\ts}{\tilde{s}}
\newcommand{\txB}{\tilde{x}_{\cB}}
\newcommand{\txN}{\tilde{x}_{\cN}}
\newcommand{\tsB}{\tilde{s}_{\cB}}
\newcommand{\tsN}{\tilde{s}_{\cN}}

% \newcommand{\Range}{\mbox{\rm range}\,}
\newcommand{\Null}{\mbox{\rm null}\,}
\newcommand{\dist}{\mbox{\rm dist}\,}
\newcommand{\Ker}{\mbox{\rm ker}\,}
\newcommand{\Rank}{\mbox{\rm rank}\,}

\newcommand{\CB}{c_{\cB}}
\newcommand{\CN}{c_{\cN}}
\newcommand{\AB}{A_{\cB}}
\newcommand{\AN}{A_{\cN}}
\newcommand{\AJ}{A_{\cJ}}
\newcommand{\DB}{D_{\cB}}
\newcommand{\DN}{D_{\cN}}
\newcommand{\csubB}{c_{\cB}}
\newcommand{\csubN}{c_{\cN}}
\newcommand{\uB}{\hat{u}_{\cB}}
\newcommand{\uN}{\hat{u}_{\cN}}

\newcommand{\widebar}{\overline}
\newcommand{\lamdba}{\lambda}

\newcommand{\logeps}{\log \frac{1}{\epsilon}}

\newcommand{\defeq}{\stackrel{\rm def}{=}}
\def\eqnok#1{(\ref{#1})}
\def\remark{\par{\it Remark}. \ignorespaces}
\def\remarks{\par{\it Remarks}. \ignorespaces}

\newcommand{\btab}{\begin{tabbing}
\ \ \= thenn \= thenn \= thenn \= thenn \= thenn \= \kill}
\newcommand{\etab}{\end{tabbing}}
\newcommand{\bu}{{\bf u}}
\newcommand{\dbu}{\delta_{\bf u}}
\newcommand{\op}{\, \makebox{\rm op} \,}
\newcommand{\eps}{\epsilon}
\newcommand{\emax}{\makebox{\bf emax }}
\newcommand{\emin}{\makebox{\bf emin }}
\newcommand{\svmax}{\makebox{\bf svmax }}
\newcommand{\svmin}{\makebox{\bf svmin }}
\newcommand{\SIG}{\sigma}

\def\stripe{\ifvmode\else\par\vskip-\baselineskip\vskip12pt\fi
\hbox to \hsize{\leaders\hrule height 4.5pt\hfil}\vskip-\baselineskip
            \vskip 3.5pt
            \hbox to \hsize{\leaders\hrule height 1.5pt\hfil}
            \vskip -10pt}

% begin slide with a heading and stripe
\def\bs#1{\begin{slide}
          \begin{center}
          {\bf #1}
          \end{center}
          \stripe}

% just begin slide
\newcommand{\jbs}{\begin{slide}}

% end slide
\newcommand{\es}{\end{slide}}

%\theoremstyle{break}
% {\theorembodyfont{\normalfont} \newtheorem{Exa}{Example}[chapter]
%                               \newtheorem{Alg}{Algorithm}[chapter]
%                               \newtheorem{Fra}[Alg]{Framework}
%                               \newtheorem{Pro}[Alg]{Procedure}
% }
